One-Dimensional Hurwitz Spaces, Modular Curves, and Real Forms of Belyi Meromorphic Functions

Hurwitz spaces are spaces of pairs (S,f) where S is a Riemann surface and f:S→ℂ^ a meromorphic function. In this work, we study 1-dimensional Hurwitz spaces ℋDp of meromorphic p-fold functions with four branched points, three of them fixed; the corresponding monodromy representation over each branched point is a product of (p−1)/2 transpositions and the monodromy group is the dihedral group Dp. We prove that the completion ℋDp¯ of the Hurwitz space ℋDp is uniformized by a non-nomal index p+1 subgroup of a triangular group with signature (0;[p,p,p]). We also establish the relation of the meromorphic covers with elliptic functions and show that ℋDp is a quotient of the upper half plane by the modular group Γ(2)∩Γ0(p). Finally, we study the real forms of the Belyi projection ℋDp¯→ℂ^ and show that there are two nonbicoformal equivalent such real forms which are topologically conjugated